Traffic cop or truth detector? The rough reception given Matt Lauer for his back-to-back interviews with Hillary Clinton and Donald Trump laid bare a disagreement over whether journalists who moderate presidential debates should call candidates out for telling lies.

Online critics hit Lauer for spending too much time on Clinton's email server and trying to cut her answers short during Wednesday's NBC forum on national security issues. The bulk of the attacks, however, centered on Lauer's failure to challenge Trump on the Republican's assertion that he opposed the war in Iraq from its beginning, despite evidence of him supporting the war in a 2002 interview.

"Everyone, and I mean everyone, knew this would happen," tweeted Paul Krugman, columnist for The New York Times. "And Matt Lauer didn't have a followup planned?"

The NBC "Today" show host wasn't talking on Thursday, laying low on a day the hashtag "Lauering the Bar" trended on Twitter. He made one knowing reference while interviewing Dana Carvey on "Today." The comic, impersonating Russian leader Vladimir Putin, commended Lauer for his work at the forum. "You have a fan," he said.

"One," replied Lauer, holding up his index finger.

The response illustrated the pressure moderators will face during the three upcoming presidential debates, starting with Lauer's NBC colleague Lester Holt on Sept. 26. Even before Wednesday's forum, which was watched by 14.7 million people, the role of moderators as fact-checkers was being talked about after Fox News' Chris Wallace, who will be in charge of the third presidential debate on Oct. 19, said in an interview that "I don't believe my job is to be a truth squad."

Others consider that a dereliction of duty.

"By not adjudicating, the moderator leaves the viewing public with a 'he said, she said' situation when the journalist picked to be onstage could say, decisively, who is right," wrote Michael Calderone of The Huffington Post.

Experts point out the difficulties of doing that in real time, particularly in a hyper-partisan atmosphere. CNN's Candy Crowley infuriated Republicans during a 2012 presidential debate for contradicting a Mitt Romney statement — precisely what Lauer was criticized for not doing.

"I don't think fact-checking is the function of the moderator," said Jim Lehrer, who moderated every first presidential debate of the general election campaign between 1988 and 2012. "It is the moderator's job to make sure the candidate has the opportunity to do the fact-checking. It's a subtle difference. If the moderator fact-checked all the time, you'd never get through it."

This week's forum was an interview, not a debate with two candidates onstage at the same time, increasing the responsibility of the questioner to challenge untruths. Since he wasn't talking, it's impossible to know whether Lauer had considered whether it would have been repetitive given that earlier in the hour, Clinton had specifically pointed out that Trump had been fudging the record on Iraq.

Emphasizing the importance of getting at the truth, Fox News Channel on Thursday afternoon had this headline onscreen beneath Trump as he talked about the forum: "Trump: I opposed the Iraq War from the beginning."

Kathleen Hall Jamieson, director of the Annenberg Public Policy Center at the University of Pennsylvania, said she believed Trump's Iraq statement was important enough for Lauer to correct. But during a debate, a moderator has many things to consider: fairness, equal time, keeping the debate moving and not getting mired in minutiae.

"You've got to make sure you're doing it to both candidates with the same set of rules," Jamieson said. "Most moderators, in the stress of the situation, don't have the capacity."

CNBC panelist Becky Quick learned the pitfalls in a GOP forum last fall when Trump pushed back against her question about something the candidate had said previously. Challenged, she didn't immediately have the back-up material available, enabling Trump to sail past the moment.

Trump's unusual campaign places even more pressure on moderators who try for fairness because he tells more falsehoods than most politicians, said Glenn Kessler, who writes the Fact Checker column for The Washington Post. Through July, Kessler had investigated the truth of 52 claims by Trump and awarded "four Pinocchios" to 33 of them, meaning they were "whoppers." By contrast, he found 14 percent of Clinton statements he studied reached that level, which he said was typical of most politicians.

Many intense partisans don't believe — or don't care — when an independent analyst questions the truth of a candidate's statements.

One way to insert fact-checking into a debate is for moderators to include it in questions posed, and there's often video evidence to illustrate contradictory statements. Lauer also could have pointed out Thursday that Trump's statement about Iraq had been fact-checked and voters could investigate it for themselves, said Alan Schroeder, author of "Presidential Debates: 50 Years of High-Risk TV."

So much is a judgment call in the moment. Even Wallace, who said fact-checking wasn't his job, during a primary campaign debate challenged a Trump claim of potential cost savings with a graphic and admonition: "Your numbers don't add up, sir."

With so many partisan, critical eyes watching, moderating can be a thankless job — one that hasn't been fully defined, Lehrer said.

"It is impossible to moderate anything without being criticized," said Lehrer, who did 12 presidential and vice presidential debates. "There's no such thing as a perfectly moderated debate. It's just not possible."